Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 2 / d / Solution

For a geometrically thin isothermal ideal-gas atmosphere, set $g=G{M}_p/R_p^2$ andHydrostatic balance gives $P(z)=P_0{e}^{-z/H}$. The observed transit photosphere at $P_{\mathrm{phot}}$ is consequently at the isothermal pressure-level transit radiusThis assumes $P_{\mathrm{phot}}<P_0$, constant $T_0$, composition, and gravity, and an opacity that selects the stated pressure.